Apportionment methods and the Liu–Layland problem

The Liu–Layland periodic scheduling problem can be solved by the house monotone quota methods of apportionment. This paper shows that staying within the quota is necessary for any apportionment divisor method to solve this problem. As a consequence no divisor method, or equivalently no population monotone method, solves the Liu–Layland problem.     

Cauchy–Gelfand problem and the inverse problem for a first-order quasilinear equation

Gelfand’s problem on the large time asymptotics of the solution of the Cauchy problem for a first-order quasilinear equation with initial conditions of the Riemann type is considered. Exact asymptotics in the Cauchy–Gelfand problem are obtained and the initial data parameters responsible for the localization of shock waves are described on the basis of the vanishing viscosity method with uniform estimates without the a priori monotonicity assumption for the initial data.     

Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers

The present paper seeks to continue the analysis in Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] on tangential boundary stabilization of Navier–Stokes equations, d=2,3$d=2,3$, as deduced from well-posedness and stability properties of the corresponding linearized equations. It intends to complement [V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] on two levels: (i) by casting the Riccati-based results of Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] for d=2,3$d=2,3$ in an abstract setting, thus extracting the key relevant features, so that the resulting framework may be applicable also to other stabilizing boundary feedback operators, as well as to other parabolic-like equations of fluid dynamics; (ii) by including, in the case d=2$d=2$ this time, also the low-level gain counterpart of the results in Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] with both Riccati-based and spectral-based (tangential) feedback controllers. This way, new local boundary stabilization results of Navier–Stokes equations are obtained over [V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear].     

Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system

We consider a parabolic–hyperbolic coupled system of two partial differential equations (PDEs), which governs fluid–structure interactions, and which features a suitable boundary dissipation term at the interface between the two media. The coupled system consists of Stokes flow coupled to the Lamé system of dynamic elasticity, with the respective dynamics being coupled on a boundary interface, where dissipation is introduced. Such a system is semigroup well-posed on the natural finite energy space (Avalos and Triggiani in Discr Contin Dynam Sys, to appear). Here we prove that, moreover, such semigroup is uniformly (exponentially) stable in the corresponding operator norm, with no geometrical conditions imposed on the boundary interface. This result complements the strong stability properties of the undamped case (Avalos and Triggiani in Discr Contin Dynam Sys, to appear). R. Triggiani’s research was partially supported by National Science Foundation under grant DMS-0104305 and by the Army Research Office under grant DAAD19-02-1-0179.     

Two-component generalized Ragnisco–Tu equation and the Riemann–Hilbert problem

Theoretical and Mathematical Physics - Using the Riemann–Hilbert approach, we investigate the two-component generalized Ragnisco–Tu equation. The modified equation is integrable in the...     

Sigma–delta quantization errors and the traveling salesman problem

In transmission, storaging and coding of digital signals we frequently perform A/D conversion using quantization. In this paper we study the maximal and mean square errors as a result of quantization. We focus on the sigma–delta modulation quantization scheme in the finite frame expansion setting. We show that this problem is related to the classical Traveling Salesman Problem (TSP) in the Euclidean space. It is known [Benedetto et al., Sigma–delta () quantization and finite frames, IEEE Trans. Inform. Theory 52, 1990–2005 (2006)] that the error bounds from the sigma–delta scheme depends on the ordering of the frame elements. By examining a priori bounds for the Euclidean TSP we show that error bounds in the sigma–delta scheme is superior to those from the pulse code modulation (PCM) scheme in general. We also give a recursive algorithm for finding an ordering of the frame elements that will lead to good maximal error and mean square error. Supported in part by the National Science Foundation grant DMS-0139261.     

On the Sylvester–Gallai and the orchard problem for pseudoline arrangements

We study a non-trivial extreme case of the orchard problem for 12 pseudolines and we provide a complete classification of pseudoline arrangements having 19 triple points and 9 double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of Böröczky. Since Melchior’s inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Grünbaum’s problems. We formulate some open problems which involve our new examples of line arrangements.     

Stability and stabilization by output feedback control of positive Takagi–Sugeno fuzzy discrete-time systems with multiple delays

This paper deals with the problem of stabilization by output feedback control of Takagi–Sugeno (T–S) fuzzy discrete-time systems with a fixed delay by linear programming (LP) and cone complementarity while imposing positivity in closed-loop. The stabilization conditions are derived using the single Lyapunov–Krasovskii Functional (LKF). An example of a real plant is studied to show the advantages of the design procedure.     

On the Cauchy–Gelfand problem

The problem posed by Gelfand on the asymptotic behavior (in time) of solutions to the Cauchy problem for a first-order quasilinear equation with Riemann-type initial conditions is considered. By applying the vanishing viscosity method with uniform estimates, exact asymptotic expansions in the Cauchy–Gelfand problem are obtained without a priori assuming the monotonicity of the initial data, and the initial-data parameters responsible for the localization of shock waves are described.     

State-space exact linearization and stabilization with feedback control in SODE economic models

The paper presents a non-conventional control engineering strategy proposed by experimental physicists, employed for controlling Dynamical Systems and basically designed for the control of nonlinear systems (Liqun and Yan Zhu in Appl. Math. Mech. 19:67–73, 1998; Liqun and Yan Zhu in Phys. Lett. A 262:350–354, 1999). After a brief presentation of the strategy—called state space exact linearization method, this is applied to design a nonlinear feedback control law as well as a modified version of this law, to control the Kaldor (Chang and Smyth in Rev. Econ. Stud. 38:37–44, 1971) and the Bonhoeffer-Van Der Pohl (Grassman in Environment, Economics and their Mathematical Models, 1994) nonlinear systems used in macro-economic business cycles.     

From the Schrödinger problem to the Monge–Kantorovich problem

The aim of this article is to show that the Monge–Kantorovich problem is the limit, when a fluctuation parameter tends down to zero, of a sequence of entropy minimization problems, the so-called Schrödinger problems. We prove the convergence of the entropic optimal values to the optimal transport cost as the fluctuations decrease to zero, and we also show that the cluster points of the entropic minimizers are optimal transport plans. We investigate the dynamic versions of these problems by considering random paths and describe the connections between the dynamic and static problems. The proofs are essentially based on convex and functional analysis. We also need specific properties of Γ-convergence which we didn?t find in the literature; these Γ-convergence results which are interesting in their own right are also proved.     

Hilbert function spaces and the Nevanlinna–Pick problem on the polydisc II

In [19], a geometric procedure for constructing a Nevanlinna–Pick problem on Dⁿ${\mathbb{D}}^n$ with a specified set of uniqueness was established. In this sequel we conjecture a necessary and a sufficient condition for a Nevanlinna–Pick problem on D²${\mathbb{D}}^2$ to have a unique solution. We use the results of [19] and Bezout?s theorem to establish three special cases of this conjecture.     

Vorticity–velocity-pressure and stream function-vorticity formulations for the Stokes problem

We study the Stokes problem of incompressible fluid dynamics in two and three-dimension spaces, for general bounded domains with smooth boundary. We use the vorticity–velocity-pressure formulation and introduce a new Hilbert space for the vorticity. We develop an abstract mixed formulation that gives a precise variational frame and conducts to a well-posed Stokes problem involving a new velocity–vorticity boundary condition. In the particular case of simply connected bidimensional domains with homogeneous boundary conditions, the link with the classical stream function-vorticity formulation is completely described, and we show that the vorticity–velocity-pressure formulation is a natural mathematical extension of the previous one.     

Characterization of the Palais–Smale sequences for the conformal Dirac–Einstein problem and applications

In this paper we study the Palais–Smale sequences of the conformal Dirac–Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the existence of infinitely many solutions to the problem provided that the underlying manifold exhibits certain symmetries.     

On the Waring–Goldbach problem for seventh and higher powers

We apply recent progress on Vinogradov’s mean value theorem to improve bounds for the function H(k) in the Waring–Goldbach problem. We obtain new results for all exponents \(k\ge 7\), and in particular establish that for large k one has

$$\begin{aligned} H(k)\le (4k-2)\log k-(2\log 2-1)k-3. \end{aligned}$$

     

On the inverse scattering problem and the low regularity solutions for the Dullin–Gottwald–Holm equation

We present an algorithm for the inverse scattering problem associated to the Dullin–Gottwald–Holm equation, arising in the study of the unidirectional propagation of waves in shallow water. In addition, a sufficient condition which guarantees the existence of the low regularity solutions for the generalized Dullin–Gottwald–Holm equation is studied by the method of energy estimation.